Difficulty: Easy
Correct Answer: 11010101
Explanation:
Introduction / Context:Converting from decimal to binary is fundamental in digital systems. The idea is to express the same quantity using powers of 2. There are two common manual methods: repeated subtraction of powers of 2 and repeated division by 2. We will apply the powers-of-2 method to convert 213 to binary cleanly and verify it afterward.
Given Data / Assumptions:
Concept / Approach:
Find the largest power of 2 ≤ number, set that bit, subtract, and continue. Powers of 2 near 213 are 128 (2^7), 64 (2^6), 32 (2^5), 16 (2^4), 8 (2^3), 4 (2^2), 2 (2^1), 1 (2^0). Set a bit to 1 when using that power, else 0.
Step-by-Step Solution:
Start: 213 ≥ 128 → set bit 7 = 1; remainder = 213 - 128 = 85.85 ≥ 64 → set bit 6 = 1; remainder = 21.21 ≥ 32 → no; bit 5 = 0.21 ≥ 16 → set bit 4 = 1; remainder = 5.5 ≥ 8 → no; bit 3 = 0.5 ≥ 4 → set bit 2 = 1; remainder = 1.1 ≥ 2 → no; bit 1 = 0.1 ≥ 1 → set bit 0 = 1; remainder = 0.Concatenate bits 7..0 → 11010101.Verification / Alternative check:
Evaluate 11010101 as decimal: 128 + 64 + 16 + 4 + 1 = 213. Match confirmed.
Why Other Options Are Wrong:
11001101 = 128 + 64 + 8 + 4 + 1 = 205.
01111001 = 64 + 32 + 16 + 8 + 1 = 121.
11100011 = 128 + 64 + 32 + 2 + 1 = 227.
Common Pitfalls:
Forgetting a power, reversing bit order, or assuming fixed-width without matching the option formatting. Always re-sum to double-check the binary pattern.
Final Answer:
11010101
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